For several types of information relations, the induced rough sets system RS does not form a lattice but only a partially ordered set. However, by studying its Dedekind-MacNeille completion DM(RS), one may reveal new important properties of rough set structures. Building upon D. Umadevi's work on describing joins and meets in DM(RS), we previously investigated pseudo-Kleene algebras defined on DM(RS) for reflexive relations. This paper delves deeper into the order-theoretic properties of DM(RS) in the context of reflexive relations. We describe the completely join-irreducible elements of DM(RS) and characterize when DM(RS) is a spatial completely distributive lattice. We show that even in the case of a non-transitive reflexive relation, DM(RS) can form a Nelson algebra, a property generally associated with quasiorders. We introduce a novel concept, the core of a relational neighborhood, and use it to provide a necessary and sufficient condition for DM(RS) to determine a Nelson algebra.

翻译：对于多种类型的信息关系，其诱导的粗糙集系统 RS 并不构成格，而仅为一个偏序集。然而，通过研究其戴德金-麦克尼尔完备化 DM(RS)，可以揭示粗糙集结构的新重要性质。基于 D. Umadevi 关于描述 DM(RS) 中并运算与交运算的工作，我们先前研究了在自反关系下定义于 DM(RS) 上的伪克林代数。本文更深入地探讨了在自反关系背景下 DM(RS) 的序论性质。我们描述了 DM(RS) 的完全并不可约元，并刻画了 DM(RS) 何时构成空间完全分配格。我们证明，即使在非传递的自反关系情形下，DM(RS) 也能构成一个通常与拟序相关联的尼尔森代数。我们引入了一个新概念——关系邻域的核心，并利用它给出了 DM(RS) 确定一个尼尔森代数的充分必要条件。